This is a take home test for calculus 3! I’m having trouble on it, and I have no idea where to start! I’m willling to pay! First come first serve! So please let me know when you have gotten all or some done! Thanks!
Harold Washington College
Math
2
09
Test 2
Name: Date: 2/2
1
/2012
(1) Show that the curvature of the helix r(t) =< a cos t,a sin t,bt > with a,b 6= 0 is
a
a2 + b2
, and also that the unit tangent vector of this curve makes a constant angle
with the z-axis whose value is θ = cos−
1
b
√
a2 + b2
.
(2) Show that the representation of r(s) =<
s +
√
s2 + 1
2
,
1
2(s +
√
s2 + 1)
,
√
2(ln(s +
√
s2 + 1)
2
>
is the arc length parametrization of the curve. In other words, show that
∥∥∥∥dr(s)ds
∥∥∥∥ = 1.
Hint. Define u = s +
√
s2 + 1 and apply the chain rule.
(
3
) Show that the parametrization of the curve r(t) =< et cos t,et sin t,et > with
respect to the arc length measured from the point t = 0 in the direction of increasing
t is r(s) =
(
s +
√
3
√
3
)
< sin
(
ln(
s +
√
3
√
3
)
)
, cos
(
ln(
s +
√
3
√
3
)
)
, 1 >
(
4
) Show that the curve r(t) =< t, 1 + t
t
,
1 − t2
t
> lies in a plane.
(5) Show that the curvature along the curve r(t) =< t− sin t, 1 − cos t, t > is
|κ| =
√
1 + 4 sin4 t
2√
(1 + 4 sin2 t
2
)3
1
(6) Show that the torsion along the curve r(t) =< t− sin t, 1 − cos t, t > is
τ = −
1
1 + 4 sin4 t
2
(7) For the curve r(t) =< 3t−t3, 3t2, 3t+t3 > use the formulas |κ| =
‖r′ (t) ×r′′ (t)‖
‖r′ (t)‖3
and τ =
|r′ (t) ×r′′ (t) ·r′′′ (t) |
‖r′ (t) ×r′′ (t)‖2
to show that |κ| = τ =
1
3(1 + t2)2
.
(8) Show that the torsion of the helix r(t) =< a cos t,a sin t,bt > with a > 0 and
b 6= 0 is
b
a2 + b2
.
(9) (a) Show that the the equation of the osculating plane for the curve r(t) =<
t,t2, t3 > is 3x− 3y + z = 1.
(10) For the helix r(t) =< a cos t,a sin t,bt > with a,b 6= 0 and t = π
3
, find the
following.
(a) The tangent vector, the normal vector, and the binormal vector.
(b)The tangent line, the normal line, and the binormal line.
(c) The rectifying plane, the normal plane, and the osculating plane.
(11) Two particles travel along the space curves r1(t) =< t,t 2, t3 > and
r2(t) =< 1 + 2t, 1 + 6t, 1 + 14t >. Do the particles collide? where? Do their paths
intersect? where?
2
(12) Find the intersection point of the helix r1(t) =< cos t, sin t, t > and the curve
r2(t) =< (1 + t), t 2, t3 >, and find the angle of intersection of these curves at that
point.
(13) Evaluate the following limit
lim
t→0
⟨
sin 10t
sin 3t
, csc t− cot t,
et −e−t − 2t
t− sin t
⟩
=
(14) At what point do the curves
r1(t) =
⟨
t, 1 − t, 3 + t2
⟩
and r2(s) =
⟨
3 −s,s− 2,s2
⟩
intersect?
(15) For the space curve r(t) =< t +
t3
3
, t−
t3
3
, t2 > determine (a) the unit tangent
vector, (b) the curvature, (c) the principal normal, (d) the binormal vector, (e) the
torsion.
(16) Find the length of the curve r(t) =< 3 cos t, 4 cos t, 5 cos t > from t = 0
to t = 2π.
(17) Evaluate the following integrals
(a)
∫
< sin(ln x),
√
t2 − 9
t3
,
t2 − t + 6
t3 + 3t
> dt =
(b)
∫
<
1√
(1 + t2) ln(x +
√
1 + t2)
,
t−
√
tan−1 t
1 + 4t2
,
1
√
t(1 + t)
> dt =
3
(18) If r(t) =< 3x2(1 + x)
4
3
4
−
9x(1 + x)
7
3
1
4
+
27(1 + x)
10
3
140
,
(2×2 − 1)
√
x2 + 1
3×3
,
2(3ax− 2b)
√
(ax + b)3
15a2
>, then
r
′
(t) =< x2(1 + x)
1
3 ,
1
x4
√
x2 + 1
, x
√
ax + b >.
(19) If r(t) =< x2
4
+
x sin 2x
4
+
cos 2x
8
,
sin3 x
3
−
sin5 x
5
, x3 cos x−3×2 sin x−6x cos x+
6 sin x >, then
r
′
(t) =< x cos2 x, sin2 x cos3 x, −x3 sin x >.
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